Difference between revisions of "2019 AIME I Problems/Problem 12"
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==Problem 12== | ==Problem 12== | ||
Given <math>f(z) = z^2-19z</math>, there are complex numbers <math>z</math> with the property that <math>z</math>, <math>f(z)</math>, and <math>f(f(z))</math> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math> and <math>n</math> such that one such value of <math>z</math> is <math>m+\sqrt{n}+11i</math>. Find <math>m+n</math>. | Given <math>f(z) = z^2-19z</math>, there are complex numbers <math>z</math> with the property that <math>z</math>, <math>f(z)</math>, and <math>f(f(z))</math> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math> and <math>n</math> such that one such value of <math>z</math> is <math>m+\sqrt{n}+11i</math>. Find <math>m+n</math>. | ||
Revision as of 04:08, 15 March 2019
Problem 12
Given 
, there are complex numbers 
with the property that , 
, and 
are the vertices of a right triangle in the complex plane with a right angle at . There are positive integers 
and 
such that one such value of is 
. Find 
.
Solution
We will use the fact that segments 
and 
are perpendicular in the complex plane if and only if 
. To prove this, when dividing two complex numbers you subtract the angle of one from the other, and if the two are perpendicular, subtracting these angles will yield an imaginary number with no real part.
Now to apply this:
![\[\frac{f(z)-z}{f(f(z))-f(z)}\in i\mathbb{R}\]](http://latex.artofproblemsolving.com/2/9/c/29cc2a2aaaea3af0c1b35260acb9cf8f70d1bfa7.png)
![\[\frac{z^2-19z-z}{(z^2-19z)^2-19(z^2-19z)-(z^2-19z)}\]](http://latex.artofproblemsolving.com/5/d/f/5df1b0913ffd02aa4cecd359fa193b132b939815.png)
![\[\frac{z^2-20z}{z^4-38z^3+341z^2+380z}\]](http://latex.artofproblemsolving.com/6/5/d/65d5d7f85b95e70c0c61506d14ee697c77cc7f9d.png)
![\[\frac{z(z-20)}{z(z+1)(z-19)(z-20)}\]](http://latex.artofproblemsolving.com/3/d/4/3d4a3da47b5ce7b3757ed6cd86d1155a6a8259e3.png)
![\[\frac{1}{(z+1)(z-19)}\in i\mathbb{R}\]](http://latex.artofproblemsolving.com/4/7/3/473e1763fc256f757a00a3edd31c6f15075b035b.png)
The factorization of the nasty denominator above is made easier with the intuition that 
must be a divisor for the problem to lead anywhere. Now we know 
so using the fact that the imaginary part of is 
and calling the real part r,
![\[(r+1+11i)(r-19+11i)\in i\mathbb{R}\]](http://latex.artofproblemsolving.com/9/4/e/94e627b14d85e87e791c0426d744adf8f850dcff.png)
![\[r^2-18r-140=0\]](http://latex.artofproblemsolving.com/4/3/5/435e71966ab02af7d5e47c37b4524cda7c6f9173.png)
solving the above quadratic yields 
so our answer is 
See Also
| 2019 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
